Given: $q = \frac{3 \pm \sqrt{1 - 3k}}{k - 4}$ Determine for which value(s) of $k$ will $q$; 2.1.1 Have equal roots 2.1.2 Be undefined Given the equation: $4x^2 + 3x + p = 0$ 2.2 Complete the following statement: 2.2.1 If the roots are non-real, then the value of the discriminant is .. - NSC Technical Mathematics - Question 2 - 2023 - Paper 1

The expression for $q$ will be undefined when the denominator equals zero. Hence, we solve:

$k - 4 = 0$

This leads to: $k = 4$

Thus, the value of $k$ for which $q$ is undefined is $k = 4$.

If the roots are non-real, the value of the discriminant must be less than zero. Therefore, we can state:

The discriminant $\Delta$ is negative.

For the equation $4x^2 + 3x + p = 0$, we use the discriminant formula:

$\Delta = b^2 - 4ac$ Here, $a = 4$, $b = 3$, and $c = p$.
Thus, the discriminant becomes:

$\Delta = (3)^2 - 4(4)(p) = 9 - 16p$

For the roots to be non-real, we require: $9 - 16p < 0$ Solving this inequality gives: $-16p < -9 \implies p > \frac{9}{16}$

In conclusion, the roots will be non-real when $p > \frac{9}{16}$.